On the stability of kalman-bucy diffusion processes

Publication Type:
Journal Article
Citation:
SIAM Journal on Control and Optimization, 2017, 55 (6), pp. 4015 - 4047
Issue Date:
2017-01-01
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© 2017 Society for Industrial and Applied Mathematics. The Kalman-Bucy filter is the optimal state estimator for an Ornstein-Uhlenbeck diffusion given that the system is partially observed via a linear diffusion-type (noisy) sensor. Under Gaussian assumptions, it provides a finite-dimensional exact implementation of the optimal Bayes filter. It is generally the only such finite-dimensional exact instance of the Bayes filter for continuous state-space models. Consequently, this filter has been studied extensively in the literature since the seminal 1961 paper of Kalman and Bucy. The purpose of this work is to review, re-prove and refine existing results concerning the dynamical properties of the Kalman-Bucy filter so far as they pertain to filter stability and convergence. The associated differential matrix Riccati equation is a focal point of this study with a number of bounds, convergence, and eigenvalue inequalities rigorously proven. New results are also given in the form of exponential and comparison inequalities for both the filter and the Riccati ow.
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